On isomorphisms between Siegel modular threefolds

The present paper is motivated by Mukai's "Igusa quartic and Steiner surfaces". There he proved that the Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. This compactification can be viewed as a Siegel modular threefold. The aim of this paper is to show that this result can be extended to other modular threefolds. The main tools are Siegel modular forms. Indeed, the construction of the degree 8 endomorphism on suitable modular threefolds is done via an isomorphism of graded rings of modular forms. By studying the action of the Fricke involution one gets a further extension of the result to other modular threefolds. The possibility of a similar situation in higher dimensions is discussed at the end of the paper.